What is the effective interest rate on a lease for a $170,000 truck with $0 down, $4,000 monthly payment over 5 years, no buy-out option at the end of the lease and $40,000 truck FMV after 5 years?

To calculate the **effective interest rate** (also known as the **implicit interest rate**) on the lease, we need to treat the lease as a form of a loan and then solve for the rate that equates the present value of the payments to the cost of the truck, minus its residual value (FMV) after 5 years.

Here's the step-by-step breakdown:

1. **Lease Amount (Present Value):**
   The truck costs $170,000, but after 5 years, its residual value (FMV) is $40,000. Effectively, you're leasing the depreciation of the truck, which is:
   \[
   \text{Depreciation} = 170,000 - 40,000 = 130,000
   \]
   So, the effective lease amount to finance is $130,000.

2. **Monthly Payment:**
   You are making monthly payments of $4,000 for 5 years (or 60 months). Thus, the total number of payments is:
   \[
   N = 5 \times 12 = 60 \text{ payments}
   \]

3. **Residual Value:**
   The truck's residual value at the end of the lease is $40,000, but since there is no buy-out option, this value affects how we calculate the lease rate.

4. **Formula for Present Value of Annuity:**
   To find the effective interest rate, we solve for the rate $ r $ in the present value annuity formula:
   \[
   130,000 = 4,000 \times \left( \frac{1 - (1 + r)^{-60}}{r} \right)
   \]
   This equation can be solved for $ r $, which represents the **effective monthly interest rate**.

Once we find the monthly interest rate $ r $, we can convert it to an **annual effective interest rate** by using the following formula:
   \[
   \text{Annual Effective Interest Rate} = (1 + r)^{12} - 1
   \]

Let me calculate this now.The **effective annual interest rate** on the lease is approximately **31.10%**.

Would you like more details on how this was calculated, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. How does the residual value of the truck affect the lease payments?
2. What is the difference between nominal and effective interest rates?
3. How does an increase in monthly payments impact the effective interest rate?
4. What would the interest rate be if there was a down payment?
5. How would the presence of a buy-out option affect the lease calculation?

**Tip:** Always differentiate between nominal and effective interest rates, as they represent different compounding frequencies.

Discover how to calculate the effective interest rate on a lease for a $170,000 truck with $4,000 monthly payments over 5 years. This problem involves key finance concepts such as present value of annuity, lease depreciation, and effective interest rate calculations. Learn how the residual value of the truck impacts the overall leasing cost.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation